Multi-variable operations

ABSTRACT

Operation of a multi-variable drilling-rig is carried out within an envelope defined by convex hulls (TC,BC) that are derived from sets of historical values for the variables accumulated in a store ( 3 ) from previous operations. A display unit ( 5 ) shows the real-time values (Q 01 –Q 10 ) of the variables on parallel axes (X 01 –X 10 ) together with upper and lower limits (Up 01 –Up 10 , Lp 01 –Lp 10 ) of the viable ranges for each variable to remain within the envelope, taking into account the current value of each other variable. The adjustment of the value of a variable (Q 03 ) required to correct for a variable (Q 08 ) found to be outside its viable range, is determined tentatively before implementation, by adjusting that value in the display to bring about re-calculation and display of the changes in viable ranges that would result from such adjustment.

This application is a National Stage Completion of PCT/BG03/02216 filedMay 22, 2003 which in turn claims priority from British PatentApplication Serial No. 0211836.2 filed May 23, 2002.

FIELD OF THE INVENTION

This invention relates to multi-variable operations.

The invention is particularly concerned with methods and systems ofoperating a controllable multi-variable process.

SUMMARY OF THE INVENTION

According to one aspect of the present invention a method of operating acontrollable multi-variable process, comprises deriving amulti-dimensional display representation of the variables according toindividual coordinate axes, defining bounds or envelopes for prospectiveoperation of the process, said bounds or envelopes being defined inaccordance with sets of values for the process-variables accumulatedrespectively from previous multiple operations of the process,indicating current values of the process-variables on their respectiveaxes of the display representation, calculating for each variableaccording to the current values of the other variables a viable range ofvalues for that individual variable consistent with operation of theprocess within said bounds or envelopes, demarcating the calculatedviable ranges of the process-variables on their respective axes of thedisplay representation, detecting the condition in which the currentvalue of any said variable is outside the viable range calculated forthat variable, and responding to said condition to adjust the value ofone or more of the variables towards maintaining the current values ofall said variables within their respective viable ranges.

The definition of the bounds or envelopes for prospective operation ofthe process may be carried out, for example, by reference to convexhulls calculated for respective pairs of the variables using the valuesaccumulated for them from the sets of previous multiple operations ofthe process. A convex hull in orthogonal coordinates is a closed polygonthat encloses all relevant data points of the two-dimensional space,whereas in parallel coordinates it is a pair of spaced linear curvesthat as between corresponding parallel axes bound the region occupied bythe lines that represent (in the parallel-coordinate space) those datapoints. A feature of convex hulls used in the present invention is thatwhen the value of one variable is fixed a range of values from maximumto minimum of the others can be derived.

The invention may be applied to assist the monitoring and optimisationof processes such as the drilling or boring of holes or shafts in theearth, and in this respect may, for example, be applied in the oil andwater industries in connection with the mining of oil, gas and water.More especially, the invention is applicable to ensuring safe andefficient operation of the process, in particular by providing amulti-dimensional display representation in which current values of someor all of the relevant variables are indicated on individual coordinateaxes together with the limits which are applicable to them formaintaining operation of the process within defined envelopes based onprevious operations of the process. The relevant limits may be indicatedin demarcation of the calculated viable ranges of the variables, so thatan operator may take appropriate action to adjust the value of one ormore of the variables to correct deviation of the value of any of thevariables outside the indicated limits.

The operator may be assisted in taking appropriate action by a facilitythat allows the operator to make tentative changes to the values of oneor more of the variables. More especially, adjustment of the value ofone or more of the variables towards maintaining the current values ofall said variables within their respective viable ranges, may include apreliminary step of tentative adjustment within the displayrepresentation of the values of the one or more variables from thecurrent values thereof and calculation of the viable ranges that wouldapply to the respective variables if that adjustment were made.

According to another aspect of the present invention a system for use inthe operation of a controllable multi-variable process, comprises meansfor deriving a multi-dimensional display representation of the variablesaccording to individual coordinate axes, means for defining bounds orenvelopes for prospective operation of the process, said bounds orenvelopes being defined in accordance with sets of values for theprocess-variables accumulated respectively from previous multipleoperations of the process, means responsive to input signals inaccordance with current values of the process-variables for indicatingthe current values of the variables on their respective axes of thedisplay representation, calculating means for calculating, for eachvariable according to the current values of the other variables, aviable range of values for that individual variable consistent withoperation of the process within said bounds or envelopes, the calculatedviable ranges of the process-variables being demarcated on theirrespective axes of the display representation, and means operable fordefining tentative change from the current values of a selected one ormore of the variables, said calculating means being responsive to thetentative change so defined for calculating the viable ranges asaforesaid that would be applicable in the event of implementation ofsaid change, for demarcation on the respective axes.

Where the invention is applied to assist drilling operations, theadvantage of existing drilling-process measurements can be taken toprovide the well-drill operator with new information in a form whichwill guide him/her to achieve improved and faster hole-drilling. Theoperator is most effectively and efficiently afforded the experience ofearlier drillings, such that his/her performance in directing thedrilling process can be greatly enhanced. In this regard, the operatormay be assisted in the time-varying and depth-dependent drilling task,by updating the envelope of normal operation from the most-recentdrilling activity either periodically or when an alarm-activity monitorindicates that conditions have changed (for example in geologicalconditions at the bottom of the hole). The accumulated sets of valuesused for defining the bounds or envelopes of the individual variables,may be changed during the course of operation for other sets of valuesaccumulated respectively from the previous multiple operations of theprocess. Different modes of drilling, use of different drill bits,insertion of a drill-string into a hole or withdrawal of it, may behandled individually with their own respective operating envelopes.

BRIEF DESCRIPTION OF THE DRAWINGS

A method and system according to the present invention will now bedescribed, by way of example, with reference to the accompanyingdrawings, in which:

FIG. 1 is a schematic representation of a system according to theinvention for use on an oil-drilling rig for detecting, advising andresponding to changes occurring during drilling;

FIG. 2 is illustrative of a display representation in multi-dimensionalspace, provided in the system of FIG. 1;

FIG. 3 is illustrative of the display representation of FIG. 2 in asubsequent phase of operation of the system of FIG. 1 leading to analarm condition in which the value of one of the process-variables liesoutside a viable range calculated for it;

FIG. 4 is illustrative (with part omitted) of the display representationof FIG. as changed in consequence of tentative action to correct for thealarm condition; and

FIG. 5 is illustrative of an alternative form of display representationto that shown in FIG. 2. for use in the system of FIG. 1: and FIG. 6 isillustrative of yet another alternative form of display representationto that shown in FIG. 2, for use in the system of FIG. 1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An example of the method and system to be described enables adrilling-rig operator to drill faster and for longer. In particular, itenables the operator to maintain the best possible rate of increase ofbore depth without leading to an unstable hole or damage to the drillbit, and to avoid unplanned interruptions to drilling arising fromevents such as pack-off (partial hole collapse). These benefits areachieved by deriving the current values of drilling variables from thereal-time data-collection system of the rig, and displaying to theoperator continuously in respect of each variable, the leeway for changeof value that is available consistent with maintaining acceptable or‘good’ drilling. If any of the current values are not in accordance with‘good’ drilling, the operator can obtain guidance on the changes to thevariables under his control, required by way of correction. The operatoris also warned through the display of the likely onset of an undesirableevent (such as pack-off), so that corrective and/or remedial action canbe taken.

In the context of description of the present specific example, it willbe assumed that there are twelve variables involved in the drillingoperation (the particular variables concerned, and the number of them,used in any implementation of the invention is subject to selection andmay clearly be different from what is described in this example). Ofthese, three real-time variables forming a first group, are directlycontrollable by the operator, being rotational speed (revolutions perminute) of the drill-bit, mud-flow rate and hook load (used to adjustthe weight on the drill-bit). Seven real-time variables forming a secondgroup are not controllable directly, being torque on the drill string,weight on the drill-bit, back pressure on the mud-pump, gas level, holedepth, change of hole depth since last measurement, and rate ofprogress. The other two variables, forming a third group, aremud-circulating density and mud-solids content; their values aredetermined after the event, being dependent on the outcome of using thevalues of the other two groups of variables.

Referring to FIG. 1, values of the twelve variables of the three groupsare supplied from the drilling-rig instrumentation 1, and are allsampled together periodically by a data collection unit 2 for writinginto a store 3. The sample-values of the first and second real-timegroups of variables are written into the store 3 marked according to thetime of sampling and in sequence with the values of other samplings ofthose same ten variables. The values of the third group of variables areacquired by an analysis unit (not shown) and written into the store 3 inassociation with the samples from the first and second groups to whichthey relate. The store 3 accordingly accumulates a combination of valuesfor the twelve variables applicable to each sampling or data point.

As drilling progresses, a multiplicity of combinations of values of thetwelve variables are accumulated in sequence from successive samplings,so as to result in the build up of an historical record of data points(each of twelve dimensions) in the store 3. This record is used toassist choice of the values of the variables of the first and secondgroups appropriate to achieve successful drilling. For this purpose, theoperator selects from the historical record a subset of data points thatrelate to successful drilling operation carried out in circumstancescomparable with those of the current operation. The selection is madeaccording to criteria set by the operator and may be adjusted in keepingwith changes in the mode of drilling as the operation progresses.

The values of each variable of the selected subset, after being scaledto the range 0 to 1, are processed in a unit 4 to calculate from thedata points represented, an operations envelope for display in anelectronic display unit 5. The calculated envelope is a closed shape(desirably the smallest) in as many dimensions as there are real-timevariables (in this case ten) which encloses all the data points of theselected subset. As displayed by the unit 5, the envelope defines inrelation to the ten real-time variables, a zone within which data pointsappropriate to successful drilling can be expected to lie; this isnominally the best operating zone or ‘BOZ’.

The BOZ envelope is calculated in this example by determining the convexhulls that are applicable between the ten variables of the first andsecond groups taken in pairs, and is displayed by the unit 5 within afield of ten equally-spaced parallel axes identified respectively withthe ten real-time variables (the first and second groups). The convexhulls for the pairs of adjacent axes of the parallel-axis system, butfor no others, are shown in the display, and the current values of thefirst and second groups of variables (namely of those variables that arecontrollable directly and those that are not) are plotted on theirrespective axes. In the latter respect, the current values of thevariables of the first and second groups are passed from the datacollection unit 2 via a unit 6 for supply to the display unit 5.

An example of the form of display representation provided by the unit 5is illustrated in FIG. 2.

Referring to FIG. 2, the current values Q01–Q10 of the ten variables ofthe first and second groups are plotted on ten equally-spaced, parallelaxes X01–X10 respectively after they have been scaled to the range 0to 1. The actual (non-scaled) values are shown digitally in graphicboxes 7 at the bottoms of those axes within the display representation.For convenience, the first three axes, X01–X03, are used for thevariables of the first group, namely, those whose values Q01–Q03 aredirectly controllable, though it is not essential for thedirectly-controllable variables to be either first or consecutive.

The convex hulls applicable between the variables of adjacent axesX01–X10 are the only ones to be included in the display. Each convexhull has two parts defining upper and lower limiting boundaries of theBOZ envelope intermediate the two axes. The upper and lower hulls of thesuccessive pairs of adjacent axes are limited to the value-range 0 to 1,and so join together to define top and bottom boundaries or chains TCand BC respectively. The display of the convex hulls, in particular ofthe chains TC and BC, is optional.

Each plot of the current value Q01–Q10 on its respective axis X01–X10,is accompanied by plots of the range within which that variable mustremain if operation is to be kept within the BOZ. The relevant range foreach variable is calculated in the unit 6 taking into account thecurrent value of each other variable, and the upper and lower limits ofthe range are represented digitally, non-scaled in the box 7 of therespective axis as well by plots of carets on the axis itself. Moreparticularly in the latter respect, the upper limits for variables ofaxes X1–X10 are represented by downwardly-directed carets Up01–Up10respectively, and the lower limits by upwardly-directed carets Lp01–Lp10respectively. For clarity, the upper carets Up01–Up10 are interconnectedin the display by a polygonal line UL and the lower carets Lp01–Lp10 bya polygonal line LL.

As referred to above, the box 7 at the bottom of the axis X01–X10 ofeach process-variable contains the real-time digital value (‘Actual’) ofthat variable together with the digital values of the upper and lowerlimits (‘UpLim’ and ‘LoLim’ respectively) of its viable range. Therepresentations within the boxes 7 at the bottoms of axes X01, X03 andX08 in the display of FIG. 2 are, for example, as given in TABLE I

TABLE I Axis X01 Axis X03 Axis X08 None None None UpLim UpLim UpLim168.73 263.34 4191.3 Actual Actual Actual 106.76 218.76 3196.0 LoLimLoLim LoLim  96.917 187.01 3169.5

The display representation provided by the unit 5 changes as the currentvalues of the variables change during drilling. Change of value of anyof the represented variables results in a corresponding change inlocation along the relevant axis X01–X10 of the respective value Q01–Q10and corresponding change of the ‘Actual’ value in the associated box 7.In general, this change will be accompanied by relocation of the caretsUp01–Up10 and Lp01–Lp10 on the axes X01–X10, and consequent changedconfigurations of the lines UL and LL, together with changes in the‘UpLim’ and ‘LoLim’ values of the box 7. The unit 5 therefore providesthe operator continuously with confirmation of the current operatingcondition, and indication of whether the current values of the variablesare within the respective ranges appropriate to ‘good’ operation.

In the display illustrated in FIG. 2, the current-value representationsQ01–Q10 are all within their respective ranges. The fact in each casethat neither limit is violated is signified not only by the location ofthe representative plot, but is confirmed by ‘None’ in the associatedbox 7 (see Table I).

FIG. 3 shows a changed condition from that of FIG. 2, in which thecurrent value Q08 of the variable represented by the axis X08 has, owingto changes in the current values of the other variables, fallen belowthe lower limit represented by the caret Lp08. The unit 6 is responsiveto conditions of this nature not only to signal an alarm state (by meansnot shown), but to indicate which limit has been violated in the box 7to the respective axis X08, and to emphasise the relevant caret (Lp08 inthis example) by enlargement and colour change. The representationswithin the boxes 7 at the bottoms of axes X01, X03 and X08 in thedisplay of FIG. 3 are, for example, as given in TABLE II

TABLE II Axis X01 Axis X03 Axis X08 None None Lower UpLim UpLim UpLim168.16 269.52 4143.3 Actual Actual Actual 104.81 266.08 3308.0 LoLimLoLim LoLim  94.671 235.14 3378.5

The fact that the lower limit of axis X08 is violated is confirmed bythe change from ‘None’ to ‘Lower’ in the box 7 to that axis.

As a general matter, change of value of any of the variables will resultin re-definition of the upper and lower limits applicable to others ofthe variables, with consequential change of location of the limitsUp01–Up10 and Lp01–Lp10 and lines UL and LL. Thus, the location of thevalue Q08 below that of the caret Lp08, can arise as in the example ofFIG. 3, in consequence of change of any of the other values Q01–Q07 andQ09–Q10, without any change having occurred in the value Q08 itself,just as well as by change of that value.

Action to rectify the condition illustrated in FIG. 3 can be taken bychanging the values of one or more of the controllable values Q01–Q03.Assistance with deciding which of the variables to change and to whatextent is provided in the display. In this regard, the unit 5 acting inconjunction with unit 6, includes provision for simulating the effect ofchanging any one or more of the represented values Q01–Q03. Moreespecially, a mouse or other pointing device (not shown) is used toidentify the current value to be changed and to draw out from it (withchange of colour) an image for location in a position along the relevantaxis, appropriate to the tentative change of value selected.

Unit 6 then re-calculates the upper and lower limits that would apply toall the variables resulting from the tentative change or changes ofvalue made, and generates for display new upper and lower caretsUp01′–Up10′ and Lp01′–Lp10′ and joining lines UL′ and LL′, that would beapplicable in the event of implementation of the change, for demarcationon the respective axes. As illustrated specifically in FIG. 4, the caretLp08′ and the line LL′ are superimposed (in distinctive colours) uponthe existing, current-display so that the operator can readily see whatthe outcome of making the tentative change of value or values would bein comparison with the current situation.

FIG. 4 shows for the circumstances illustrated in FIG. 3, theconsequence of tentative change of the value of the variable of axis X03from Q03 to Q03′. As illustrated, the tentatively-changed value, whichas well as being represented on the axis X03 as value Q03′ is showndigitally in an added graphic box 8, would result in reduction of thelower limit currently represented by caret Lp08 on axis X08. The lowerlimit is reduced to a value that in the superimposed, tentative display,is represented by a caret Lp08′. The caret Lp08′ is located below thecurrent value Q08 of the variable, so the change if implemented, wouldclear the alarm.

A box 8 appears in the display representation on the axis of any of thedirectly-controllable variables, in the present system the variables ofaxes X01–X03, for which tentative change has been made. In the examplerepresented in FIG. 4, the box 8 on the axis X03 gives indication of theactual, non-scaled value Q03′ as illustrated by Table III.

TABLE III Axis X03 Move to 260.92

Once the operator has determined a change of one or more of the valuesof axes X01–X03 that will clear the alarm, action is taken to implementthe change in the drilling process, in order to bring that process backwholly within the BOZ envelope. The return to this condition ismonitored by the operator and any further departure acted upon todetermine tentatively the change required to correct for it.

The system may be operable in a mode in which it is responsive to analarm condition to calculate and indicate to the operator, arecommendation of the change best suited for implementation incorrecting the alarm condition.

As drilling progresses it can be expected that the rate at which alarmconditions arise will remain reasonably constant, or rise and fallgently, for some time and then show a sharp rise. A sharp rise indicatesthat drilling conditions (for example, geological conditions) havechanged from those for which the BOZ was selected. At this point a newBOZ is constructed by selection of a new sub-set of data accumulated inthe store 3; the new sub-set will include values of the real-timevariables of the first and second groups, but may also include values ofthe first group if they are available. The sub-set selected will be thataccumulated during the most-recent interval of drilling, say drillingthroughout the last 10,000 feet. Any data points associated withabnormal events (for example a stuck bit) are eliminated from thesub-set before it is used to construct the new BOZ.

The decision as to when a new BOZ is to be utilised and the selection ofthe data for its construction and the updating of the systemaccordingly, may be carried out by the drilling engineer. However, theoperator of the display system may identify the need for a new BOZ andinitiate automated selection of the relevant data, construction of thenew BOZ from it, and updating of the system. Alternatively, the systemmay operate entirely automatically to initiate, select, construct andupdate the system with a new BOZ.

Although the multi-dimensional display representation provided by theunit 5 has been illustrated and described above in the context ofvertical, parallel axes, the unit 5 may be arranged to provide itselectively in this vertical form or with the axes horizontal.Furthermore, the unit 5 may also, or alternatively, be arranged toprovide the display representation in a polar form with radial axes. Apolar-form of display representation is illustrated in FIG. 5.

Referring to FIG. 5 twelve axes P01–P12 are in this case used, and theexistence of an alarm condition in which the current value Q08 of thevariable of axis P08 violates the lower limit Lp08 of that axis, isillustrated.

A further alternative form of display is illustrated in FIG. 6. In this,each of the real-time or on-line variables V1–V4, is represented by ahorizontal bar (the bars might instead be vertical). The current valuesQ1–Q4 of the variables V1–V4 are each indicated by a symbol spaced alongthe bar according to its magnitude; a dot is used for the symbol in thiscase but an open-centred circle may be used as an alternative.Furthermore, according to this example, a triangular symbol is added tothe dot symbol in each case where the variable is directly controllable,and upper and lower limits U and L respectively, of the variables areindicated by cross-bars. In the example illustrated, the upper limit U2of the variable V2 is violated.

When the operator moves the current-value symbol of one of thecontrollable variables for tentative investigation of the effect of thechange on the limits of the other variables, the triangular symbol ofthe moved variable remains fixed in the location appropriate to itscurrent-value. Thus, in the case illustrated where there is tentativechange of the value of the variable V4, to Q4′, the triangular symbolfor that variable remains in the location appropriate to thecurrent-value Q4. The effect on the limits for each variable V1 and V2is illustrated in the example of FIG. 6, by distinctive upper-limitcross-bars L1′ and L2′ and distinctive lower-limit cross-bars U1′ andU2′; the change would have the desirable effect of correcting thecondition in which the value Q2 of the variable V2 violates the upperlimit U2.

1. A method of operating a controllable multi-variable well- orshaft-drilling process, comprising steps of; deriving amulti-dimensional display representation of drilling process variablesof the drilling process according to individual coordinate axes;defining one of bounds or envelopes for prospective operation of thedrilling process, said bounds or envelopes being defined in accordancewith sets of values for drilling process variables accumulatedrespectively from previous multiple operations of the drilling process;indicating current values of the drilling process variables onrespective axes of the display representation, the current values of thedrilling process variables comprising real-time values of the drillingprocess variables, and one or more of the drilling process variablesbeing drilling process variables that are directly controllable inperformance of the drilling operation; calculating for each drillingprocess variable according to current values of other drilling processvariables, a viable range of values for that individual drilling processvariable consistent with operation of the drilling process within saidbounds or envelopes; demarcating the calculated viable ranges of thedrilling process variables on their respective axes of the displayrepresentation; detecting a condition in which the current value of anysaid drilling process variable is outside a viable range calculated forthat drilling process variable; responding to said condition to adjustthe value of one or more of the directly controllable drilling processvariables towards maintaining the current values of all said variableswithin their respective viable ranges; and wherein the accumulated setsof values used for defining the one of bounds or envelopes, are changedduring course of operation of the process for other sets of values forthe drilling process variables accumulated respectively from theprevious multiple operations of the drilling process, accumulated setsof values used for defining the one of bounds or envelopes being changedfor other sets of values in dependence upon the rate at which thecondition is detected.
 2. The method according to claim 1 wherein thebounds or envelopes are convex hulls calculated for respective pairs ofthe drilling process variables using the values accumulated for them insaid sets.
 3. The method according to claim 2 wherein the convex hullsfor the pairs of drilling process variables of adjacent axes of themulti-dimensional display representation are indicated in the displayrepresentation.
 4. The method according to claim 1 wherein the axes ofthe multi-dimensional display representation are parallel to oneanother.
 5. The method according to claim 1 wherein the axes of themulti-dimensional display representation are angularly spaced from oneanother.
 6. The method according to claim 1 wherein the demarcation ofthe calculated viable range for each drilling process variable is byindicators of upper and lower values respectively of the viable range onthe axis of that drilling process variable in the displayrepresentation.
 7. The method according to claim 6 wherein adjustment ofthe value of one or more of the directly controllable drilling processvariables towards maintaining the current values of all said drillingprocess variables within their respective viable ranges, includes apreliminary step of tentative adjustment within the displayrepresentation of the values of the one or more drilling processvariables from the current values thereof and calculation of the viableranges that would apply to the respective drilling process variables ifthat adjustment were made.
 8. The method according to claim 7 whereinthe viable range that would apply to each drilling process variable ifthe adjustment were made is demarcated on an axis of that drillingprocess variable in the display representation.
 9. A system for use inoperation of a controllable multi-variable well- or shaft-drillingprocess, comprising means for deriving a multi-dimensional displayrepresentation of drilling process variables of the process according toindividual coordinate axes, means for defining one of bounds orenvelopes for prospective operation of the drilling process, said one ofbounds or envelopes being defined in accordance with sets of values fordrilling process variables accumulated respectively from previousmultiple operations of the process, means responsive to input signals inaccordance with current values of the drilling process variables forindicating current values of the drilling process variables onrespective axes of the display representation, the current values of thedrilling process variables being real-time values of the drillingprocess variables and at least some of the drilling process variablesbeing drilling process variables that are directly controllable inperformance of the drilling process, calculating means for calculating,for each drilling process variable according to the current values ofother drilling process variables, a viable range of values for thatindividual drilling process variable consistent with operation of thedrilling process within said bounds or envelopes, the calculated viableranges of the drilling process variables being demarcated on respectiveaxes of the display representation, and means operable for definingtentative change from the current values of a selected one or more ofthe drilling process variables, said calculating means being responsiveto the tentative change so defined for calculating the viable ranges asaforesaid that would be applicable in the event of implementation ofsaid change, for demarcation on the respective axes, and wherein thesystem further comprises means for detecting a condition in which thecurrent value of any said drilling process variable is outside theviable range calculated for the drilling process variable, and meansresponsive to the rate of occurrence of the condition for changing thesets of values defining the one of bounds or envelopes during course ofoperation of the drilling process for other sets of values for thedrilling process variables accumulated respectively from the previousmultiple operations of the drilling process.
 10. The system according toclaim 9 wherein the one of bounds or envelopes are convex hullscalculated for respective pairs of the drilling process variables usingthe values accumulated for them in said sets.
 11. The system accordingto claim 10 wherein the convex hulls for the pairs of drilling processvariables of adjacent axes of the multi-dimensional displayrepresentation are indicated in the display representation.
 12. Thesystem according to claim 9 wherein the axes of the multi-dimensionaldisplay representation are parallel to one another.
 13. The systemaccording to claim 9 wherein the axes of the multi-dimensional displayrepresentation are angularly spaced from one another.
 14. The systemaccording to claim 9 wherein the demarcation of the calculated viablerange for each drilling process variable is by indicators of upper andlower values respectively of that range on the axis of that drillingprocess variable in the display representation.